basic group theory questions

taken from D&F, question 1.1.7:

Let (i.e. ). Show that is an abelian group.

﻿The question asks for verification of the group axioms, although I'm a little confused with showing the existence of inverses. One can say that for an arbitrary , which is the identity. But notice that I said , and that's confusing me. Is ? Is it literally the negative of ? I would presume as much since I actually took that into consideration when I added . But is not part of by construction. So how do I show the existence of inverses?

Another look

I have been taking topology this semester and was thinking this same problem could be approached in a more topological way by looking at the bijection from (the unit circle) by and then this is obviously an abelian group with the operation rotations of x degrees about the origin of the complex plane and you can check to see that this would preserve addition easier this way, at least in my opinion. It is not a homeomorphism (the inverse function is not continuous at ), but I don't think that will affect this in terms of it still being and isomorphism.